Questions: Determine whether the following statement is true or false. If (B) is a subset of a universal set (U), then (B cup U=B). True False

Transcript text: Determine whether the following statement is true or false. If $B$ is a subset of a universal set $U$, then $B \cup U=B$. True False

Solution

Solution Steps

To determine whether the statement "If $ B $ is a subset of a universal set $ U $, then $ B \cup U = B $" is true or false, we need to understand the properties of set operations. Specifically, we need to check if the union of a subset $ B $ with its universal set $ U $ results in the subset $ B $ itself.

Step 1: Define the Sets $ B $ and $ U $

Given: \[ B = \{1, 2, 3\} \] \[ U = \{1, 2, 3, 4, 5\} \]

Step 2: Check if $ B \subseteq U $

Since every element of $ B $ is also an element of $ U $, we have: \[ B \subseteq U \]

Step 3: Calculate $ B \cup U $

The union of $ B $ and $ U $ is: \[ B \cup U = \{1, 2, 3, 4, 5\} \]

Step 4: Compare $ B \cup U $ with $ B $

We need to check if: \[ B \cup U = B \] Substituting the values, we get: \[ \{1, 2, 3, 4, 5\} \neq \{1, 2, 3\} \]